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Traveling wave solutions for an age-since addiction opioid epidemic model with distributed delays and diffusion

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  • Darazirar, Rassim
  • Djilali, Salih

Abstract

In this paper, we investigate the existence and nonexistence of traveling wave solutions for an age-structured opioid epidemic model with distributed delays. Understanding the spatial outbreak of opioid addiction is very important for anticipating epidemic invasion and informing public health interventions. Therefore, it is important to determine whether opioid addiction can invade new territories. This invasion can be analyzed by investigating the existence of traveling wave solutions. It is shown that two key factors influence the existence of such solutions: the basic reproduction number, R0, and the wave speed ζ. Indeed, when R0>1, there exists a minimal wave speed, denoted by ζ∗>0, which leads to two distinct scenarios. The first is the existence of a traveling wave solution connecting the opioid-free steady state to the positive steady state when ζ>ζ∗. The second scenario occurs when 0<ζ<ζ∗, in which case no nontrivial traveling wave solution exists that connects the two steady states. The mathematical findings are supported and validated numerically through some graphical representations.

Suggested Citation

  • Darazirar, Rassim & Djilali, Salih, 2026. "Traveling wave solutions for an age-since addiction opioid epidemic model with distributed delays and diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 250(C), pages 54-83.
    • Handle: RePEc:eee:matcom:v:250:y:2026:i:c:p:54-83
    DOI: 10.1016/j.matcom.2026.06.012
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